Let F be a given field whose elements are called scalars and V be a non-void set whose elements are called vectors.
The set V is a vector space or a linear space over the field F if the following conditions are satisfied:
For any two vectors $\alpha, \beta \in V$, $\alpha + \beta \in V$ i.e the vectors are closed under vector addition.
For any two vectors, $\alpha, \beta \in V$, $\alpha + \beta = \beta + \alpha$ i.e vector addition is commutative.
For any two vectors, $\alpha, \beta, \gamma \in V$, $\alpha + (\beta + \gamma) = (\alpha + \beta) + \gamma$ i.e vector addition is associative.
There exists a unique vector $\phi \in V$ such that $\alpha + \phi = \alpha = \phi + \alpha$ for all $\alpha \in V$ i.e there exists an additive identity in V.
For any vector $\alpha \in V$, there exists a unique vector $\omega \in V$ such that $\alpha + \omega = \phi$ i.e there exists additive inverse for each element in V.